Idempotents of the Hecke algebra become Schur functions in the skein of the annulus

Abstract

The Hecke algebra H n contains well known idempotents E λ which are indexed by Young diagrams with n cells. They were originally described by Gyoja [5].A skein theoretical description of E λ was given by Aiston and Morton [2]. The closure of E λ becomes an element Q λ of the skein of the annulus. In this skein, they are known to obey the same multiplication rule as the symmetric Schur functions s λ as stated in theorem 8.2. But previous proofs of this fact as in [1] used results about quantum groups which were far beyond the scope of skein theory.Our elementary proof was motivated by [6] and uses only skein theory and basic algebra. The skein of a planar surfaceWe consider a planar surface F with designated n incoming and n outgoing boundary points for some integer n ≥ 0. The Homfly skein S(F ) is defined as the module of linear combinations of oriented tangles in F quotiented by regular isotopy (i.e. Reidemeister moves II and III), the two local skein relations in figure 1 where v and s are variables, and the relation that a disjoint simple closed curve can be removed from a diagram at the expense of multiplication with the scalar δ =

Geometrical relations and plethysms in the Homfly skein of the annulus

“…The element h m ∈ C m , which is taken to represent the complete symmetric function of degree m, is the closure of the element (1/α m )a m ∈ H m , where a m = π∈Sm s l(π) ω π is one of the two basic quasi-idempotent elements of H m . Here ω π is the positive permutation braid associated to the permutation π ∈ S m with length l(π), and α m is given by the equation a m a m = α m a m (see [2,8,10]). Using the other quasi-idempotent b m = π∈Sm (−s) −l(π) ω π in a similar way determines the element e m which represents the elementary symmetric function.…”

“…In general each coefficient in h m is a rational function with denominator [m]!. As a further example, the coefficient θ (3,3,2) [8] .…”

“…. , λ l ) (see [5,8]). Since h i = h i [10, Lemma 3.7], the elements Q λ are not affected by the mirror map.…”

“…Section 5 introduces the meridian maps and the skein-theoretic representatives {Q λ } for the Schur functions {s λ }, following Lukac [8]. In Theorem 15, we expand equation (4.2) using symmetric functions to express A m in the basis {Q λ } of Schur functions.…”

The elliptic Hall algebra and the deformed Khovanov Heisenberg category

“…We recall a useful result of Lukac in [Luk05]. There is a natural closure map cl n : Tr(Br n ) → Sk Proof.…”